Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13026 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,14,19,29,38,30,39,16,21,13,18,15,20,31,40)(12,22,24,34,32,37,27,17,26,33,23,41,25,36,28,35);(6,7,8,9)(10,12)(11,17)(13,27)(14,28)(15,32)(16,26)(18,36)(19,37)(20,41)(21,35)(22,39)(23,30)(24,31)(25,29)(33,40)(34,38);(10,12)(11,17)(13,28)(14,27)(15,32)(16,26)(18,37)(19,36)(20,41)(21,35)(22,40)(23,31)(24,30)(25,29)(33,39)(34,38)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6014);